Chapter 3 Gaussian process emulation for probabilistic global
3.3 Li-ion battery model
3.4.2 Sensitivity analysis
3.6 3.8 4 4.2 Cell voltage / V 0 10 20 30 Time/s 3.4 3.6 3.8 4 4.2 Cell voltage / V
Figure 3.2: Example predictions y⇤r,j (dashed lines) of the cell voltage during the discharge-charge cycle yj⇤ (solid lines) using r = 5 for m = 50 (left) and r = 10 for m = 100 (right). The worst case predictions (highest ✏⇤) are the thick lines and 4 further examples are shown for each value ofm.
Figure 3.1 shows Tukey boxplots of the relative errors on the test set {(⇠⇠⇠⇤
j,y⇤j)}mj=1t using the training set {(⇠⇠⇠j,yj)}mj=1 with m = 50 and m = 100 training points for an increasing number of PCA basis dimensions r. The errors were defined as follows✏⇤ =||y⇤
r,j y⇤j||/||yj⇤||, in whichy⇤r,j is the mean
GP prediction of y⇤j using Eq. (3.19). It can be seen that the emulator error decreases with increasing r, plateauing at r = 5 for m = 50 and r = 10 for m= 100. It is also evident that increasing the number of training points leads to more accurate predictions. For both values of m, example predictions of the cell voltage during the discharge-charge cycle are shown in Figure 3.2. The worst case predictions (highest✏⇤) are shown, alongside 4 further examples for each value of m. It is clear that both values of m capture the trends well, and are quantitatively accurate, even in the worst case. For m = 100, the predictions are particularly accurate, so for the SAm = 100 and r = 10 were selected.
3.4.2
Sensitivity analysis
The SA was performed using the SAFE package developed by Pianosi et al. [128]. For the variance-based method a uniform distribution was placed on the factors and pointsX = [⇠i,j],i= 1, . . . ,2k,j = 1, . . . , N (N = 5000) in the 2k
hypercube using a Latin hypercube design were sampled. The physical ranges were 0.1✏p 0.4, 0.5Rp[µm] 2 and 0.4SOCin 0.6, and the factors
were scaled to obtain X = [0,1]3. The sampled inputs were used to produce the three input matrices A 2 RN⇥k, B 2 RN⇥k and C
i 2 RN⇥k, i = 1, . . . , k,
from which the QoI values qA = f(A), qB = f(B) and qCi = f(Ci) were
extracted. TheSi and STi were then calculated from Eqs. (3.11).
ϵ p Rp SOCin 0 0.2 0.4 0.6 0.8 1 Sensitivity main effects total effects
Figure 3.3: Box plots of the main and total e↵ects for the energy efficiency.
ϵ p Rp SOCin 0 0.2 0.4 0.6 0.8 1 Sensitivity main effects total effects
Figure 3.4: Box plots of the main and total e↵ects for the cell voltage drop during discharge.
Figure 3.3 presents both the main and total e↵ects for the three factors. As expected, the particle size and porosity are the most influential, while the initial SOC mainly a↵ects the open-circuit potential (slight shift in the charge- discharge curve up or down) so has relatively little influence onq. The porosity determines the e↵ective ionic conductivity (the volume fraction of electrolyte is ✏p) and since the ohmic loss is predominantly su↵ered in the ionic phase,
rate depends upon the concentration (per unit volume of the electrode) of Li+ according to the Butler-Volmer law (3.30), so a restricted supply of Li+ in the positive electrode will lead to a large concentration overpotential for a fixed current (the overpotential in (3.30) must increase ascdecreases in order to maintain a fixed left hand side, i.e, applied current density). The particle radius determines the level of mass transport resistance for the solid Li (which has to di↵use through the particle to react at R =Rp) as well as the specific
surface area for reaction (smaller particles lead to higher specific areas). Thus, increasing the particle radius will lead to a higher concentration overpotential and, therefore, a deterioration in performance.
The ordering here is specifically for the energy efficiency (for a constant current charge-discharge cycle the energy efficiency is simply the average cell voltage during discharge divided by the average cell voltage during charge) so it is dangerous to draw too many conclusions. For another QoI, such as the voltage drop during discharge, ✏p has the greatest influence, followed by Rp
and lastly SOCin, as shown in Figure 3.4. The combined e↵ect of an increased
Ohmic drop and a higher concentration overpotential on the total polariza- tion caused by a lower ✏p outweighs the e↵ect of an increased concentration overpotential caused by a smaller Rp.
0 0.1 0.2 0.3 0.4 Mean of EEs 0 0.05 0.1 0.15 0.2 0.25
Standard deviation of EEs
Rp ϵ p SOCin 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Mean of EEs 0 0.05 0.1 0.15 0.2 0.25 0.3
Standard deviation of EEs
Rp
ϵ p
SOCin
Figure 3.5: Means and standard deviations of the elementary e↵ects with and without confidence intervals in the case of the energy efficiency.
400 800 1200 1600 2000 No of model evaluations 0 0.1 0.2 0.3 0.4 0.5 Mean of EEs Rp ϵ p SOCin 400 800 1200 1600 2000 No of model evaluations 0 0.1 0.2 0.3 0.4 0.5 Mean of EEs Rp ϵ p SOCin
Figure 3.6: Convergence of the elementary e↵ects for di↵erent numbers of model evaluations without confidence bounds (left) and with 95% confidence intervals (right) in the case of the energy efficiency.
The next results are for the elementary e↵ect test (EET) on the same data set. A uniform distribution was selected for the three factors, which were again scaled to yield X = [0,1]3. A major di↵erence between the variance- based method and the EET is the sampling strategy. The EET is highly efficient, requiring only M(k+ 1) model evaluationsvs. N(k+ 2) to calculate the main e↵ect indices; in the results above, N(k + 2) = 5000⇥(3 + 2) = 25000, much higher than typical values of M(k + 1). Figure 3.5 shows the mean and standard deviation estimates using M = 100 trajectories (100 ⇥ (3 + 1) = 400 model runs), both with and without confidence bounds. The confidence bounds were obtained using bootstrapping [129], which consists of re-sampling the base points with replacement to produce P copies of the trajectories and for each of the P copies to use the EET to estimate µi and i. This provides empirical distributions overµi and i from which means and
confidence intervals (CIs) (such as the 95% CIs in Figure 3.5) can be estimated. The ranking of the inputs is the same as in the variance-based method. The trends in the means of theµi both with and without confidence intervals
are depicted in Figure 3.6 for increasing M. The means stabilize at around M = 500 (2000 model runs). There are small but noticeable fluctuations in the mean for ✏p around the value 0.25 even at much higher values of M but this behaviour is stable. The cause is the broad range of ✏p in comparison with the other factors and, therefore, the relatively small number of samples.
Moving to a higher value of M the confidence intervals shrink, suggesting greater accuracy in the predictions. The ranking, however, is accurate even for very low numbers of M, which shows that the EET is more efficient than the variance based method.
Further indication of this is provided in Figure 3.7, which shows the EET predictions for di↵erentM in the case of the voltage drop during charge. The results are again consistent with the variance-based method (there are similar fluctuations in the mean for ✏p). Although time cost is not an issue for the emulator, which provides extremely rapid predictions (on the order of a few seconds for 2000 predictions), in cases where a full simulator is used the much lower number of model runs for the EET represents an enormous advantage. 400 800 1200 1600 2000 No of model evaluations 0 0.05 0.1 0.15 0.2 0.25 0.3 Mean of EEs ϵ p Rp SOCin 400 800 1200 1600 2000 No of model evaluations 0 0.05 0.1 0.15 0.2 0.25 0.3 Mean of EEs ϵ p Rp SOCin
Figure 3.7: Convergence of the elementary e↵ects for di↵erent numbers of model evaluations without confidence bounds (left) and with 95% confidence intervals (right) in the case of the voltage drop during discharge.
To investigate the sensitivity of the charge-discharge curve, the main and total e↵ects of the PCA coefficients wi(⇠⇠⇠) are examined using the same
Latin hypercube design and N = 5000 (e.g., ⌘E is replaced with wi(⇠⇠⇠), i 2
{1, . . . , r}). The results are depicted in Figure 3.8 for w1 and w2. Figure 3.9 shows an example (from the test set) of the contributions from the PCA eigen- vectors (wivi, up to i = 4) towards the final (mean centred) voltage profile.
The first two contributions can be seen to have by far the most influence. The sensitivities of the coefficientsw1 and w2 are highest for ✏p and Rp, with
roughly equal contributions from each, while higher-order coefficients (w3 and w4) were more heavily influenced by SOCin.
p Rp SOCin 0 0.2 0.4 0.6 0.8 1 Sensitivity 1st coefficient main effects total effects p Rp SOCin 0 0.2 0.4 0.6 0.8 1 Sensitivity 2nd coefficient main effects total effects
Figure 3.8: Main and total e↵ects for the first two PCA coefficients (for the cell voltage curve).
0 10 20 30 Time/s 3.4 3.6 3.8 4 4.2 4.4
Cell voltage / V mean
1 PC 2 PCs 3 PCs 4 PCs 0 10 20 30 Time/s -0.15 -0.1 -0.05 0 0.05 0.1 Voltage contributions/V w1 v1 w2 v2 w3 v3 w4 v4
Figure 3.9: An example of the contributions from the PCA eigenvectors (wivi)
towards the final (mean centred) voltage profile. In the left-hand figure, wivi
is successively added to the mean.
3.5
Concluding remarks
SA is often computationally unfeasible with complex computer models. In such cases emulators, of varying degrees of sophistication, can be employed. Quantifying the uncertainty in the emulator predictions is desirable, but this
is only achievable for certain approaches. For multivariate outputs (especially in high dimensional spaces), SA under uncertainty is especially challenging, even when the QoI is a scalar.
In this chapter a GP emulation approach is proposed for performing SA under uncertainty when the model output is multivariate (possibly in a high-dimensional space). An example for a Li-ion battery is presented, re- vealing that the method is efficient and accurate. It is also able to perform a probabilistic SA on scalar QoIs and also on the output itself by ranking sensi- tivity measures for the random principal coefficients. Two di↵erent methods (EET and VBSA) are presented for achieving this aim, either through full MC sampling or by using semi-analytical expressions that are extensions of those derived by Oakley and O’Hagan [116]. The EET needs less number of model evaluations to provide accurate results as it belongs to OAT sam- pling method, while VBSA needs more number of model evaluations in order to extract higher order statistics. It also shown in this chapter that despite the more model evaluations VBSA usually needs, adapting the GPE method developed here this limitation can be overcome.